Problem: What is the largest integer value of $x$ for which $5-4x>17$?
Solution: Adding $4x$ to both sides, we have $5 > 17+4x$. Then, subtracting $17$ from both sides, we have $-12 > 4x$. Finally, dividing both sides by $4$, we have $-3 > x$. This inequality states that $x$ is strictly less than $-3$. The largest integer satisfying that condition is $\boxed{-4}$.